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Computers & Industrial Engineering 112 (2017) 156–174

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Computers & Industrial Engineering

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A new multi-criteria model based on interval type-2 fuzzy sets and EDASmethod for supplier evaluation and order allocation with environmentalconsiderations

http://dx.doi.org/10.1016/j.cie.2017.08.0170360-8352/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected], [email protected]

(M. Keshavarz Ghorabaee).

Mehdi Keshavarz Ghorabaee a,⇑, Maghsoud Amiri a, Edmundas Kazimieras Zavadskas b, Zenonas Turskis b,Jurgita Antucheviciene b

aDepartment of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba’i University, Tehran, IranbDepartment of Construction Technology and Management, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 March 2017Received in revised form 11 August 2017Accepted 13 August 2017Available online 16 August 2017

Keywords:Supplier evaluationEnvironmental criteriaOrder allocationMulti-criteria decision-makingInterval type-2 fuzzy setsEDAS method

Nowadays environmental performance of suppliers becomes more important because of competitiveconditions. Besides, the economic performance has been a significant factor for companies to choose theirsuppliers. In this paper, a new integrated model is proposed for supplier evaluation and order allocationwhich considers both environmental and economic factors. We use the EDAS (Evaluation based onDistance from Average Solution) method and interval type-2 fuzzy sets for evaluation of suppliers withrespect to environmental criteria. According to this evaluation two parameters are defined for each sup-plier: positive score and negative score. These parameters, together with cost parameters, are utilized topropose a multi-objective mathematical model for determination of order quantity from each supplier. Anumerical example is used in this paper to show the applicability of the proposed integrated model. Also,a sensitivity analysis is made to examine the effect of weighting environmental criteria on total purchas-ing cost and quantity of order from each supplier. The results show that the proposed model is efficientand applicable for real-world problems.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Companies are forced to improve their performance and opti-mize their business processes because of today’s competitive envi-ronment (Chiu & Chiou, 2016; Otay & Çebi, 2016). In such anenvironment, companies are confronted with many challengesfrom different perspectives like demographic changes, emergenceof disruptive technologies, digitalization and environmental issues.Demographic changes can lead to significant changes in labor mar-ket and labor costs in terms of increase or decrease in the size ofthe workforce, and accordingly it can affect quality and/or priceof products or services of companies (Du & Yang, 2014; Richter,2014). Disruptive innovations and technologies which can createnew markets and value networks are another challenge for compa-nies in a competitive environment. Companies should makedynamic strategies like a dynamic commercialization strategyand flexibility in R&D management to prevail over this challenge(Guo, Tan, Sun, Cao, & Zhang, 2016; Marx, Gans, & Hsu, 2014). Inte-

gration of digital technologies in different processes or digitaliza-tion usually increases the performance of companies. Therefore,using information technology (IT) management can help compa-nies to overcome their rivals in a competitive environment. ITcan be used to reduce costs by improving productivity and effi-ciency and increase revenues by exploiting opportunities throughexisting customers or finding new customers (Mithas & Rust,2016; Ye & Wang, 2013).

The attention of consumers, businesses and governments to theenvironmental issues has been increased in recent years. Societiesare concerned with the environmental impact of human activitieson natural resources (Toro, Franco, Echeverri, & Guimarães, 2017).Today, companies cannot ignore environmental issues because ofincreasing public awareness in environmental protection and gov-ernment regulation about these issues. If they disregard the envi-ronmental issues in their business processes, they will probablyface with serious problems in competing with other companiesin the global market (Hänninen & Karjaluoto, 2017). Accordingly,companies are learning to purchase raw materials, products andservices from suppliers that can provide them with low cost andat the same time with environmental responsibility (Lee, Kang,Hsu, & Hung, 2009; Sinha & Anand, 2017). Specifically, new envi-

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M. Keshavarz Ghorabaee et al. / Computers & Industrial Engineering 112 (2017) 156–174 157

ronmental legislations such as RoHS Directive (Restriction ofHazardous Substances Directive), WEEE Directive (Waste Electricaland Electronic Equipment Directive) and EuP (Eco-design Require-ment for Energy-using Product) increase the pressure on compa-nies to make their processes more environmentally friendly(Akman, 2015).

Evaluation of suppliers and selection of appropriate ones forprocurement of needed materials can be an important activityfor improvement and optimization of processes in many compa-nies (Valipour Parkouhi & Safaei Ghadikolaei, 2017). In supplychain management, selection of appropriate suppliers is a strategicdecision that can affect the quality and price of the final product ofa company (Chai & Ngai, 2015; Dey, Bhattacharya, Ho, & Clegg,2015; Razmi, Kazerooni, & Sangari, 2016). The current researchfocuses on the environmental aspects as a competitive advantage.

Supplier evaluation and selection problem can be considered asa multi-criteria decision-making (MCDM) problem because it usu-ally involves some alternatives that are evaluated with respectsome criteria (Ho, Xu, & Dey, 2010). Evaluation process in theMCDM problems is usually interlocked with uncertainty of infor-mation. The fuzzy sets theory is the most common tool to deal withthe uncertainty of information in many MCDM problems (Mardani,Jusoh, Zavadskas, Khalifah, & Nor, 2015; Mardani, Zavadskas,Khalifah, Jusoh, & Nor, 2016). The type-2 fuzzy sets (T2FSs) arean extension of ordinary fuzzy sets which were proposed byZadeh (1975). The T2FSs are very flexible to model the uncertaintyof information because the membership values of them are alsofuzzy sets (Keshavarz Ghorabaee, Amiri, Kazimieras Zavadskas, &Antucheviciene, 2017; Mendel, 2007). An interval type-2 fuzzyset (IT2FS) is a special case of type-2 fuzzy set which has been usedin MCDM problems by many researchers (Mohammadi, Farahani,Noroozi, & Lashgari, 2017; Olfat, Amiri, Bamdad Soufi, & Pishdar,2016).

Many methods have been proposed to handle decision-makingproblems with multiple criteria in the past years (Mardani et al.,2015). The EDAS (Evaluation based on Distance from Average Solu-tion) method is a new and efficient method which proposed byKeshavarz Ghorabaee, Zavadskas, Olfat, and Turskis (2015) andapplied to the inventory classification problem. KeshavarzGhorabaee, Zavadskas, et al. (2015) demonstrated the efficiencyof the EDAS method by comparing it with some other MCDMmethods. Fuzzy extension of the EDAS was proposed byKeshavarz Ghorabaee, Zavadskas, Amiri, and Turskis (2016) andapplied to supplier selection problem. Also, Kahraman et al.(2017) developed an intuitionistic fuzzy EDAS method and usedit for selection of solid waste disposal site. Peng and Liu (2017) pro-posed some algorithms for soft decision-making with neutrosophicsets based on the EDAS method, new similarity measure soft set.Peng, Dai, and Yuan (2017) developed some interval-valued fuzzysoft decision making methods based on MABAC (Multi-Attributive Border Approximation area Comparison), similaritymeasure and EDAS method. Stanujkic, Zavadskas, KeshavarzGhorabaee, and Turskis (2017) presented an extension of the EDASmethod based on interval grey numbers. This method has alsobeen used in some other real-world MCDM problems (Ecer,2017; Stevic, Vasiljevic, & Veskovic, 2016; Trink�uniene et al.,2017; Turskis & Juodagalviene, 2016; Zavadskas, Cavallaro,Podvezko, Ubarte, & Kaklauskas, 2017).

In this study, a new integrated model based on IT2FSs and theEDAS method is proposed for supplier evaluation and orderallocation with environmental consideration. Some steps of theEDAS method and arithmetic operations of IT2FSs are used toevaluate suppliers with respect to environmental criteria. Theoutcome of this evaluation process is two parameters for eachsupplier: positive scores and negative scores. The purchasingcosts and the parameters determined are utilized to formulate a

multi-objective linear programming for determination of the quan-tity of order from each supplier. We use a fuzzy programmingapproach to solve this multi-objective model. A sensitivity analysisis also performed to examine impact of changing weights of envi-ronmental criteria on total purchasing cost and order quantityfrom each supplier. The results show that the integrated proposedmodel is applicable to real-world problems.

The rest of this paper is organized as follows. In Section 2, wepresent a brief review of the literature on supplier evaluation withenvironmental considerations, supplier evaluation and order allo-cation and applications of IT2FSs in MCDM problems. In Section 3,the proposed methodology is described in detail. In Section 4, weapply the proposed model to an example of supplier evaluationand order allocation with environmental considerations. A sensi-tivity analysis is made in Section 5. Section 6 presents discussionand future directions. Finally, conclusions are presented inSection 7.

2. Literature review

In this section, a brief review of the recent literature is pre-sented in three sub-section including supplier evaluation withenvironmental considerations, supplier evaluation and order allo-cation and applications of IT2FSs in MCDM problems.

2.1. Supplier evaluation with environmental considerations

In recent years, environmental criteria have been used by manyresearchers in the process of evaluation of suppliers. ‘‘Green sup-plier” and ‘‘sustainable supplier” are the most common termswhich have been used in the studies of this field. Some researchershave performed reviews on green and sustainable supplier evalua-tion studies that interested readers can refer to their studies. Forexample, Govindan, Rajendran, Sarkis, and Murugesan (2015) pre-sented a review of multi criteria decision making approaches forgreen supplier evaluation and selection, and Zimmer, Fröhling,and Schultmann (2016) conducted a review of models for support-ing sustainable supplier selection, monitoring and development. Inthe following some of recent studies in this field are reviewed.

Banaeian, Mobli, Fahimnia, Nielsen, and Omid (2016) comparedthe application of technique for order preference by similarity toan ideal solution (TOPSIS), VIKOR (in Serbian: VIseKriterijumskaOptimizacija I Kompromisno Resenje) and grey relational analysis(GRA) methods in supplier selection problem. Then they appliedthese methods to a green supplier evaluation and selection studyfor an actual company from the agri-food industry in fuzzyenvironment.

Liao, Fu, and Wu (2016) proposed a new integrated fuzzyMCDM approach based on three methods including fuzzy AHP,fuzzy additive ratio assessment (ARAS) and multi-segment goalprogramming (MSGP) to handle green supplier selection problems.Their method allows decision makers to set multiple segment aspi-ration levels for green supplier selection problems. They appliedthe proposed model to a problem in a watch manufacturingcompany.

Shahryari Nia, Olfat, Esmaeili, Rostamzadeh, andAntucheviciene (2016) developed an MCDM approach based onintuitionistic fuzzy sets (IFSs) and interval-valued intuitionisticfuzzy sets (IVIFSs). They used the Choquet integral operator andfuzzy measures for evaluation of suppliers with respect to environ-mental criteria and applied their approach to a manufacturingcompany.

Govindan, Kadzinski, and Sivakumar (2016) presented a novelapproach based on PROMETHEE (Preference Ranking OrganizationMETHod for Enrichment of Evaluations) method and Simos

158 M. Keshavarz Ghorabaee et al. / Computers & Industrial Engineering 112 (2017) 156–174

procedure for evaluation of green suppliers using a group compro-mise ranking. They showed the applicability of their approach witha case study in an Indian food industry.

Qin, Liu, and Pedrycz (2017) developed an extended TODIM (anacronym in Portuguese of interactive and multi-criteria decisionmaking) based on IT2FSs for multi-criteria group decision making.They introduced a new distance measure based the a-cuts of theIT2FSs and extended TODIM method based on this distance mea-sure and prospect theory. Then they used the proposed methodfor evaluation of suppliers with respect to environmental criteria.

Mathivathanan, Kannan, and Haq (2017) examined the interre-lated influences among sustainable supply chain managementpractices with a particular look at the automotive industry. Theyproposed a framework based on the Decision Making Trial andEvaluation Laboratory (DEMATEL) method, for evaluation of auto-motive industry supply chain management practices in an emerg-ing economy like India.

Tseng, Lim, Wu, Zhou, and Bui (2017) proposed a hybrid methodto enhance green supply chain practices. They used the fuzzy Del-phi method to screen the evaluation criteria and then a convergedinterval-valued triangular fuzzy numbers-grey relation analysis(IVTFN-GRA) weighting method used to handle the uncertaintyand incomplete information with interdependence relations. Theyapplied their method to a Taiwanese electronic focal firm.

Luthra, Govindan, Kannan, Mangla, and Garg (2017) developeda framework for evaluation of sustainable suppliers by integrationof the AHP and VIKOR methods. Based on the literature andexperts’ opinions, they identified 22 criteria in economic, environ-mental, and social dimensions, which are the sustainability dimen-sions, for evaluation of supplier. The AHP method was used toweight the criteria and the VIKOR method was utilized to rankthe suppliers.

Yazdani, Chatterjee, Zavadskas, and Hashemkhani Zolfani(2017) proposed a framework to address the relationship betweencustomer requirements and evaluation criteria for green supplierselection problem. They used the DEMATEL method to constructa relationship structure, the quality function deployment (QFD)to identify degree of relationship and the complex proportionalassessment (COPRAS) method to rank the suppliers.

Sen, Datta, and Mahapatra (2017) developed a novel decisionsupport framework to deal with supplier selection problems withgreen and resiliency criteria. Their framework included adominance-based method with the fuzzy sets theory. Thedominance-based method used by them was a simplified versionof the PROMETHEE and TODIMmethods with lower computationalsteps. By using a numerical example, they compared the proposedapproach with the fuzzy TOPSIS and fuzzy VIKOR methods.

2.2. Supplier evaluation and order allocation

The supplier evaluation and order allocation problem includestwo main processes. The evaluation of suppliers is the first processof this problem, and the second process is related to allocation oforder to suppliers with respect to the evaluation result of them.This problem usually involves two stages of a supply chain. How-ever, it is very important problem because of its effects on allstages. Although there are some review papers in this field(Aissaoui, Haouari, & Hassini, 2007; Amindoust, Ahmed, &Saghafinia, 2013; Molinè & Coves, 2013; Setak, Sharifi, &Alimohammadian, 2012), a comprehensive review has not beenmade yet. Here we review some recent studies in this field.

Singh (2014) presented a hybrid approach for prioritization ofsuppliers and allocation of demand among the suppliers. Theirapproach was based on the TOPSIS method and mixed-integer lin-ear programming (MILP). The objective function of their model wasmaximization of total purchase value of items with some

constraints including demand condition, budget, lead-time ofdelivery, and supplier capacity.

Scott, Ho, Dey, and Talluri (2015) proposed an integratedmethod to handle supplier selection and order allocation problemin a stochastic environment with multiple stakeholders and multi-ple criteria. They used a combined analytic hierarchy process–quality function deployment (AHP–QFD) approach and a chanceconstrained optimization algorithm for evaluation of suppliersand allocation of optimal orders among them. They applied theproposed decision support system to the bioenergy industry.

Moghaddam (2015) developed a model for evaluation of suppli-ers and determination of the optimal quantity of refurbished partsand final products in a reverse logistics network. The proposedmodel was based on fuzzy multi-objective programming approachand AHP. The Monte Carlo simulation was used to obtain Pareto-optimal solutions of the problem and a computational study wasdone to validate the model.

Arabzad, Ghorbani, Razmi, and Shirouyehzad (2015) proposed atwo-phase approach for supplier evaluation and order allocationproblem with respect to both qualitative and quantitative criteria.The criteria were defined based on strengths, weaknesses, opportu-nities, and threats (SWOT) analysis. Then a fuzzy TOPSIS methodwas used to determine the evaluation score of suppliers. Theresults of the fuzzy TOPSIS method were used as inputs for a linearprogramming model to allocate the orders.

Çebi and Otay (2016) presented a two-stage approach for sup-plier evaluation and order allocation problem in an uncertain envi-ronment. In the first stage, a fuzzy MULTIMOORA (multi-objectiveoptimization by ratio analysis plus the full multiplicative form)method was utilized to evaluate suppliers with respect to somesubjective criteria. Then a fuzzy goal programming was used toobtain optimal order from each supplier in a multi-productproblem.

PrasannaVenkatesan and Goh (2016) introduced a newapproach for evaluation of suppliers and optimization of theirorder allocation under disruption risk. Performance score valuesof suppliers were obtained using the fuzzy AHP and fuzzy PRO-METHEE methods, and a mixed-integer linear programmingapproach was used to formulate the mathematical model. Theyapplied the particle swarm optimization (PSO) algorithm to deter-mine Pareto-optimal solutions.

Hu, Xiong, You, and Yan (2016) proposed a hybrid approach tohandle supplier evaluation and order allocation problem. Theyused the fuzzy analytic hierarchy process and mixed integer pro-gramming in their proposed approach. To show the feasibility ofthe proposed approach they applied it to supplier evaluation andorder allocation of a plastic and textile manufacturing company.

Govindan and Sivakumar (2016) proposed an integrated MCDMapproach for supplier evaluation and order allocation problem. Theevaluation of suppliers was done with respect to green criteria, andthe fuzzy TOPSIS method was used to prioritize the suppliers. Theyformulated a multi-objective MILP model to determine optimalorder from each supplier, and a weighted additive model was usedto solve it.

Kumar, Rahman, and Chan (2016) considered the dimensions ofsustainability (economic, social and environmental) in the supplierevaluation and order allocation problem. They presented anintegrated MCDM approach based on fuzzy AHP and fuzzymulti-objective linear programming to deal with this problem.The proposed approach was applied to an example of Indianautomobile company.

Hamdan and Cheaitou (2017) developed an integrated MCDMapproach for supplier evaluation and order allocation problembased on fuzzy AHP, fuzzy TOPSIS and multi-objective mathemat-ical modelling. To evaluate suppliers, they used two types of crite-ria including green and traditional criteria. The weighted global


criterion method and branch-and-cut algorithm were utilized tosolve the model.

2.3. Applications of IT2FSs in MCDM problems

As previously mentioned, interval type-2 fuzzy sets are veryefficient and flexible to handle uncertainties in comparison to ordi-nary fuzzy sets. This type of fuzzy sets has been used in MCDMproblems by many researchers in recent years. Celik, Gul, Aydin,Gumus, and Guneri (2015) performed a comprehensive reviewon the applications of IT2FSs in multi-criteria decision-makingproblems. Interested readers are referred to this article for furtherinformation. Some of the recent studies in this field are reviewed inthe following.

Kiliç and Kaya (2016) proposed a new model for city-ranking inTurkey. They simultaneously used IT2FSs and crisp sets to addressambiguities and relativities of the problem. The proposed modelwas utilized for multi-criteria decision-making process of grantsallocation, and the applicability of it was assessed by data of a realcase in the Middle Black Sea Development Agency in Turkey.

Celik and Taskin Gumus (2016) developed an outranking multi-criteria decision-making method based on interval type-2 fuzzysets to evaluate preparedness and response ability of non-governmental humanitarian relief organizations. They used inter-val type-2 fuzzy AHP to determine the weights of some critical suc-cess factors as the criteria of the problem. Then the PROMETHEE(stands for: Preference Ranking Organization METHod for Enrich-ment of Evaluations) method was utilized to evaluate some organi-zations in Turkey.

Keshavarz Ghorabaee, Zavadskas, Amiri, and Antucheviciene(2016b) introduced a new ranking method for IT2FSs and com-pared it with some other ranking methods to show the efficiencyof it. They developed a new method of assessment based on fuzzyranking and aggregated weights (AFRAW) to deal with groupdecision-making problems with IT2FSs. Subjective weightsexpressed by decision-makers and objective weights calculatedbased on a deviation-based method were used to determine theaggregated weights of criteria.

Soner, Celik, and Akyuz (2017) integrated the AHP and VIKORmethod under interval type-2 fuzzy environment. They demon-strated the proposed approach using a hatch cover design selectionproblem. In structure of bulk carrier ships, this problem is veryimportant to protect cargo from the external damages and preventwater ingress. In their approach the AHP and VIKOR methods wereused to determine weights of criteria and rank the alternatives,respectively.

Deveci, Demirel, and Ahmetoglu (2017) addressed a planningproblem of an airline company in Turkey to launch a new routeat an airport in the North American region. They proposed anMCDM approach based on the TOPSIS method and interval type-2 fuzzy sets to select a new route from five different destinations.The results of their research show the feasibility of the proposedapproach.

Baykasoglu and Gölcük (2017) developed a new interval type-2fuzzy multi-criteria decision making model based on the TOPSISand DEMATEL methods. Their model uses hierarchical decomposi-tion approach to reduce complexity of problems, DEMATEL to con-sider interdependencies among criteria and hierarchical TOPSIS torank alternatives.

Senturk, Erginel, and Binici (2017) introduced a new approachbased on the analytic network process (ANP) and interval type-2fuzzy sets for evaluation of third-party logistics providers. Theinner/outer dependencies among criteria and pairwise comparisonbetween criteria/sub-criteria were modelled in their approachusing IT2FSs.

Chen (2017) developed a prioritized interval type-2 fuzzyaggregation operator and applied it to multi-criteria decision-making with prioritized criteria. Relationship between criteriawas considered in a situation that a lack of satisfaction by higherpriority criteria cannot be compensated by lower priority criteria.The proposed method was applied to a landfill site selectionproblem.

Yu, Wang, and Wang (2017) presented a new multi-attributiveborder approximation area comparison (MABAC) method based onIT2FSs. They first introduced an algorithm to decompose IT2FSsinto embedded ordinary fuzzy numbers, and then used the algo-rithm to determine the likelihood of IT2FSs. Finally the MABACmethod was utilized to rank alternatives based on the likelihoodof IT2FSs. They applied their method to an example of selectinghotels from a tourism website.

Celik (2017) proposed a proactive multi-criteria decision-making tool to locate temporary shelters in disaster operationsmanagement. The proposed method was based on the DEMATELmethod and interval type-2 fuzzy sets. For evaluation of the causeand effect factors, 14 criteria were considered by 9 disaster opera-tion managers, and results showed practical benefits of the pro-posed method.

Çolak and Kaya (2017) presented an integrated decision-makingmodel for prioritization of renewable energy alternatives in Tur-key. Their decision-making model was based on the intervaltype-2 fuzzy AHP and hesitant fuzzy TOPSIS methods. Intervaltype-2 fuzzy AHP method was used to obtain the weights of deci-sion criteria, and hesitant fuzzy TOPSIS method was applied torank renewable energy alternatives.

Kundu, Kar, and Maiti (2017) introduced a new method basedon relative preference index and generalized credibility measureto rank interval type-2 fuzzy sets. Using the proposed rankingmethod, they developed a new method to solve fuzzy MCDM prob-lems with linguistic variables which are based on IT2FSs. Theyapplied the proposed method to a transportation mode selectionproblem and showed the efficiency of their approach.

Zhong and Yao (2017) extended an ELECTRE (stands for: ELim-ination Et Choix Traduisant la REalité) method using interval type-2 fuzzy sets. They proposed an a-based distance method to mea-sure the proximity between the interval type-2 fuzzy numbers.Then an IT2FS entropy measure and an entropy weight model weredeveloped to obtain the criteria weights without any subjectiveinformation. The feasibility and applicability of the proposedapproach were examined by using a supplier selection problem.

Görener, Ayvaz, Kus�akcı, and Altınok (2017) proposed a three-phase hybrid approach based on the IT2FSs to address the supplierevaluation problem in the aviation industry. In the first stage themost significant evaluation criteria were defined. Then they usedan interval type-2 fuzzy AHP method to determine the importanceof the criteria defined. Finally, an interval type-2 fuzzy TOPSISmethod was utilized for evaluation of the performance of suppliersin Turkish airlines.

2.4. Summary

Based on the literature, environmental criteria are very impor-tant in the evaluation process of suppliers in a supply chain. Thesupplier evaluation and order allocation problem can involveuncertainty of information. Using interval type-2 fuzzy sets canhelp us to handle higher degrees of uncertainty in multi-criteriadecision-making problem. Although some studies have considereduncertainty and environmental criteria in the supplier evaluationand order allocation problem, there has been no study in the liter-ature which uses interval type-2 fuzzy sets to deal with the uncer-tainty of this problem. Also, most of the related researches haveonly defined one score function for evaluation of suppliers. In this


study, we can define positive and negative score functions by theEDAS method to formulate a multi-objective mathematical modelfor order allocation. The structure of the proposed model can helpto capture more degrees of uncertainty and consider environmen-tal criteria as two objectives.

3. Methodology

In this section, we present the preliminaries of the proposedapproach first, and then the steps of the proposed approach aredescribed in detail.

3.1. Preliminaries

3.1.1. Interval type-2 fuzzy setsThe type-2 fuzzy sets were introduced as an extension of fuzzy

sets (Zadeh, 1975). This type of fuzzy sets can capture moredegrees of uncertainty and lead to more rational model in anuncertain environment. In the following, some of the importantdescriptions related to this type of fuzzy sets are presented.

An interval type-2 fuzzy set is defined based on two levels ofmembership functions (lðxÞ) including upper membership func-tion (UMF) and lower membership function (LMF) which formfootprint of uncertainty (Mendel, John, & Feilong, 2006). An IT2FSwhich has the UMF and LMF with trapezium (or trapezoidal)shapes is called trapezium (or trapezoidal) interval type-2 fuzzy

set (TIT2FS). Let us denote by ~~E a TIT2FS. An example of this typeof fuzzy sets is shown in Fig. 1, and the mathematical representa-tion of it can be defined as follows:

~~E ¼ ð~Etjt 2 fL;UgÞ ¼ ðet1; et2; et3; et4;hEt1;h

Et2jt 2 fL;UgÞ ð1Þ

In this representation, ~EL shows the LMF and ~EU represents the

UMF of ~~E. Moreover, hEt1 and hE

t2 denote the membership valuesof et2 and et3, respectively (t 2 fL;Ug).

Suppose that ~~E and ~~G are two TIT2FSs and d is a crisp number.The arithmetic operations of this type of fuzzy sets are defined asfollows (Chen & Lee, 2010; Keshavarz Ghorabaee, Amiri, SalehiSadaghiani, & Hassani Goodarzi, 2014; Keshavarz Ghorabaee,Amiri, Salehi Sadaghiani, & Zavadskas, 2015):

� Addition

~~E� ~~G ¼ et1 þ gt1; et2 þ gt2; et3 þ gt3; et4 þ gt4;ð

minðhEt1;h

Gt1Þ; minðhE

t2; hGt2Þjt 2 fL;Ug

�ð2Þ

~~Eþ d ¼ et1 þ d; et2 þ d; et3 þ d; et4 þ d;hEt1;h

Et2jt 2 fL;Ug

� �ð3Þ

Fig. 1. An example of TIT2FS.

� Subtraction

~~E� ~~G ¼ et1 � gt4; et2 � gt3; et3 � gt2; et4 � gt1;ð

minðhEt1;h

Gt1Þ;minðhE

t2;hGt2Þjt 2 fL;Ug

�ð4Þ

~~E� d ¼ ðet1 � d; et2 � d; et3 � d; et4 � d;hEt1; h

Et2jt 2 fL;UgÞ ð5Þ

� Multiplication

~~E� ~~G ¼ mt1;mt2;mt3;mt4;minðhEt1;h

Gt1Þ;minðhE

t2;hGt2

� �jt 2 fL;UgÞ

ð6Þ

where

mti ¼minðetigti; etigtð5�iÞ; etð5�iÞgti; etð5�iÞgtð5�iÞÞ if i ¼ 1;2

max etigti; etigtð5�iÞ; etð5�iÞgti; etð5�iÞgtð5�iÞ

� �if i ¼ 3;4

8<: ð7Þ

~~E:d ¼et1d; et2d; et3d; et4d;h

Et1; h

Et2jt 2 fL;UgÞ

�if k P 0

et4d; et3d; et2d; et1d;hEt1; h

Et2jt 2 fL;Ug

� �if k < 0

8><>: ð8Þ

Division of a TIT2FS by a crisp number l can be defined usingd ¼ 1=l and l– 0.

� Defuzzification

T ð~~EÞ ¼ 12

Xt2fL;Ug

et1 þ ð1þ hEt1Þet2 þ ð1þ hE

t2Þet3 þ et44þ hE

t1 þ hEt2

!ð9Þ

where T ð~~EÞ denote the crisp score of ~~E.

3.1.2. Ranking TIT2FSsIn this paper, an efficient method which were proposed by

Keshavarz Ghorabaee, Zavadskas, Amiri, and Antucheviciene(2016a) is used to determine ranking scores of TIT2FSs in the pro-cess of the proposed approach. We summarize this method asfollows.

This method is based on the membership area of the subtrac-

tion of two TIT2FSs. Suppose that ~~Es and~~Et are two TIT2FSs. Firstly,

the total area under the UMF and LMF of ~~Es and ~~Et is calculated.These areas are symbolized by SEU

k and SELk (k 2 fs; tg) and shown

in Fig. 2.

Secondly, the subtraction of two TIT2FSs (~~Es � ~~Et) is calculated

and denoted by ~~D. Then the areas under the UMF and LMF of ~~Dfor positive and negative domains are computed. SDp

t and SDnt

(t 2 fL;Ug) represent the areas for positive and negative domains,respectively. These parameters are depicted in Fig. 3.

According to these areas, the possibility degree of ~~Es over~~Et is

calculated as follows:

pð~~Es P~~EtÞ ¼

Wp þ ðKðWp �WnÞ � PsÞWp þWn þKðWp �WnÞ

ð10Þ

where

Wp ¼ ðhDL2 � SDp

LÞ þ ðhDU2 � SDp

UÞ ð11Þ

Wn ¼ ðhDL1 � SDn

L Þ þ ðhDU1 � SDn

UÞ ð12Þ

Ps ¼SEU

s þ SELs

SEUs þ SEL

s þ SEUt þ SEL

t

ð13Þ

Fig. 3. (a) The area under the UMF of ~~D, (b) the area under the LMF of ~~D.

Fig. 2. (a) Total area under the UMF of ~~Ek , (b) total area under the LMF of ~~Ek .


and KðxÞ is a simple function which is defined as follows:

KðxÞ ¼1 if x ¼ 00 if x– 0

�ð14Þ

In a special case, when all elements of ~~Es and ~~Et (ekt1 to ekt4,t 2 fL;Ug and k 2 fs; tg) are equal to zero, the possibility degree

of ~~Es over~~Et is assumed to be equal to 0.5 (pð~~Es P

~~EtÞ = 0.5).Suppose that we have n TIT2FSs, using the possibility degrees

and the formula presented by Xu (2001) we can calculate the rank-ing score of each TIT2FS as follows:

Rð~~EiÞ ¼1

nðn� 1ÞXnj¼1

pð~~Ei P~~EjÞ þ

n2� 1

!ð15Þ

3.1.3. The EDAS methodThe EDAS method is a new and efficient multi-criteria

decision-making method which proposed by KeshavarzGhorabaee, Zavadskas, et al. (2015). Evaluation process in thismethod is based on positive and negative distances from theaverage solution. Suppose that we have an MCDM problem withn alternatives and m criteria, and the decision-matrix is definedas follows:

X ¼

x11 x12 � � � x1j � � � x1mx21 x22 � � � x2j � � � x2m

..

. ... . .

. ... . .

. ...

xi1 xi2 � � � xij � � � xim

..

. ... . .

. ... . .

. ...

xn1 xn2 � � � xnj � � � xnm

2666666666664

3777777777775ð16Þ

where xij denote the performance value (rating) of ith alternative onjth criterion (i = 1, 2, . . . , n and j = 1, 2, . . . , m). Also, we define theweight of jth criterion by wj, where 0 < wj < 1 and

Pmj¼1wj = 1.

The steps of using this method are presented as follows:

Step 1. Determine the average solution elements (V j) withrespect to each criterion shown as follows:

V j ¼Pn

i¼1xijn

ð17Þ

Step 2. Calculate the positive distance (Pdij) and negative distance(Ndij) of each elements of the decision-matrix from the calculatedelements of the average solution using the following equations:

Pdij ¼maxð0;xij�VjÞ

Vjif j 2 BC

maxð0;Vj�xijÞVj

if j 2 NC

8<: ð18Þ

Ndij ¼maxð0;Vj�xijÞ

Vjif j 2 BC

maxð0;xij�VjÞVj

if j 2 NC

8<: ð19Þ

where BC and NC are the sets of beneficial and non-beneficial crite-ria, respectively.

Step 3. Compute the weighted summation of the calculated pos-itive and negative distances for each alternative as follows:

SPi ¼Xmj¼1

wjPdij ð20Þ

NP i ¼Xmj¼1

wjNdij ð21Þ


Step 4. Calculate the normalized values of SPi andNPi as follows:

SPðnÞi ¼ SPi

maxk

SPkð22Þ

NPðnÞi ¼ 1� NPi

maxk

NPkð23Þ

Step 5. Calculated the appraisal score of each alternative usingthe following equation:

Asi ¼12ðSPðnÞ

i þNPðnÞi Þ ð24Þ

Step 6. Rank the alternatives according to decreasing values ofAsi.

According to the steps of the EDAS method, increase in thevalues of SPi and decrease in the values of NPi lead to increas-ing the desirability of an alternative. In other words, the valuesof SPi and NPi should be maximized and minimized,respectively.

3.2. The proposed model for supplier evaluation and order allocation

In this section, a new integrated model is proposed for sup-plier evaluation and order allocation. Firstly, we use TIT2FSsand some steps of the EDAS method to evaluate suppliers anddetermine positive and negative scores of them. Then a multi-objective linear programming model is developed to allocateorders to each supplier. A fuzzy programming approach is uti-lized to solve the multi-objective model and obtain final solu-tion. The proposed model can be used to answer the followingquestions:

� Which supplier should we select for procurement of raw mate-rial with respect to environmental and economic criteria?

� How much raw material should we order from each selectedsupplier with respect to environmental and economic criteria?

� How can environmental criteria affect the order quantity ofselected suppliers?

� How can environmental criteria affect the total purchasing costof raw material?

Moreover, a company which aims to use the proposed modelshould meet the following conditions:

� The company should be able to appoint some experts asdecision-makers which are familiar with the supplier evalua-tion and order allocation problem.

� The decision-makers should be able to identify some potentialsuppliers for procurement of raw material. Also, they shouldbe agreed on the identified suppliers for evaluation process.

� The decision-makers should be able to define some environ-mental criterial for evaluation of suppliers. They should beagreed on the selected criteria and their definitions.

� The parameters of the order allocation problem like purchasingcost and average defect rate of each supplier should be definedby the decision-makers.

The steps of the proposed model are as follows:

Step 1. Formation of a group of decision-makers and identifica-tion of potential suppliers and related criteria for evaluationprocess.

Step 2. Evaluation of the importance of each criterion and alsothe performance value of each supplier on each criterion byeach decision-maker.

In this step, the elements of fuzzy decision matrix and fuzzyweights of criteria are determined with respect to each decision-maker using TIT2FSs. Suppose that we have n suppliers, m criteriaand k decision-makers in the evaluation process. Then the follow-ing equations describe these elements:

Xp ¼ ½~~xijpn�m ð25Þ

Wp ¼ ½ ~~wjp1�m ð26Þ

where ~~xijp symbolizes the performance value of ith supplier on jth

criterion given by the pth decision-maker, and ~~wjp represents theweight of jth criterion given by the pth decision-maker.

Step 3. Calculation of aggregated values for the performancevalue of each supplier on each criterion and also the weight ofeach criterion.To obtain aggregated values, the average opinion of decision-

makers is calculated as follows:

~~xij ¼1k�k

p¼1

~~xijp ð27Þ

~~wj ¼1k�k

p¼1

~~wjp ð28Þ

Step 4. Determination of the elements of average solution.As previously mentioned, the evaluation process in the EDAS

method is based on distance from the average solution. Accordingto Step 1 of the EDAS method and aggregated performance values,we can use the following equation to determine these elements:

~~V j ¼1n�n

i¼1

~~xij ð29Þ

Step 5. Calculation of the positive and negative distances fromthe average solution.In this step, we first define a function to compare a TIT2FS with

zero and determine the maximum value, shown as follows:

Zð~~EÞ ¼~~E if Tð~~EÞ > 0

0 if Tð~~EÞ 6 0

8<: ð30Þ

Then the following equations are used to calculate the positiveand negative distances:

ggPd ij ¼

Zð~~xij�~~VjÞ

T ð~~VjÞif j 2 BC

Zð~~Vj�~~xijÞT ð~~VjÞ

if j 2 NC

8>><>>: ð31Þ

ggNd ij ¼

Zð~~Vj�~~xijÞT ð~~VjÞ

if j 2 BC

Zð~~xij�~~VjÞ

T ð~~VjÞif j 2 NC

8>><>>: ð32Þ

where BC and NC are the sets of beneficial and non-beneficial crite-ria, respectively.

Step 6. Calculation of the weighted sums of positive and nega-tive distances for all suppliers, shown as follows:ggSP i ¼ �

m

j¼1ð ~~wj � ~~PdijÞ ð33Þ

ggNP i ¼ �m

j¼1ð ~~wj �

~~NdijÞ ð34Þ


Step 7. Determination of positive scores (RðggSP iÞ) and negative

scores (RðggNP iÞ) of each supplier using the ranking method pre-sented in Section 3.1.2.Step 8. Formulation of a mathematical linear programmingmodel for allocation of order among suppliers.

The following notations are used in the model:

xi
Order quantity of ith supplier yi Binary variable of allocation (yi = 1 if xi > 0, and
otherwise yi = 0)

RðggSP iÞ
Positive score of ith supplier
RðggNP iÞ
Negative score of ith supplier
CPi

Purchasing cost of ith supplier

pi
Average defect rate of ith supplier li Average rate of lateness (in delivery) of ith supplier CAPi Maximum capacity of ith supplier Qmini Minimum allowable order quantity of ith supplier Pmax Maximum allowable defected material Lmax Maximum allowable lateness DT Total demand Z1 Total positive score of supplied material Z2 Total negative score of supplied material Z3 Total purchasing cost
According to the parameters of the EDAS method and the otherparameters and variables, the following multi-objective linear pro-gramming model is proposed to determine order allocation.

Max Z1 ¼Xni¼1

RðggSP iÞxi ð35Þ

Min Z2 ¼Xni¼1

RðggNP iÞxi ð36Þ

Min Z3 ¼Xni¼1

CPi xi ð37Þ

Subject to:

Xni¼1

xi ¼ DT ð38Þ

Xni¼1

pixi 6 Pmax ð39Þ

Xni¼1

lixi 6 Lmax ð40Þ

xi P yiQmini 8i ð41Þ

xi 6 yiCAPi 8i ð42Þ

xi P 0 and yi 2 f0;1g ð43Þ

Step 9. Solving the mathematical model using the fuzzy multi-objective programming approach proposed by Zimmermann(1978).To obtain a solution for the mathematical model (Eqs. (35)–

(43)), we first define the membership functions of the objectivesas follows:

lðZ1Þ ¼0 for Z1 6 Zmin

1

1þ Z1�Zmax1

Zmax1 �Zmin

1for Zmin

1 6 Z1 6 Zmax1

1 for Zmax1 6 Z1

8>><>>: ð44Þ

lðZ2Þ ¼

0 for Zmax2 6 Z2

1� Z2�Zmin2

Zmax2 �Zmin

2for Zmin

2 6 Z2 6 Zmax2

1 for Z2 6 Zmin2

8>><>>: ð45Þ

lðZ3Þ ¼

0 for Zmax3 6 Z3

1� Z3�Zmin3

Zmax3 �Zmin

3for Zmin

3 6 Z3 6 Zmax3

1 for Z3 6 Zmin3

8>><>>: ð46Þ

Then the following model is solved.

Max k ð47Þ

Subject to:

k 6 1þ Z1 � Zmax1

Zmax1 � Zmin

1

ð48Þ

k 6 1� Z2 � Zmin2

Zmax2 � Zmin

2

ð49Þ

k 6 1� Z3 � Zmin3

Zmax3 � Zmin

3

ð50Þ

And Eqs. (38)–(43).

It should be noted that Zmaxr and Zmin

r (r = 1, 2, 3) denote the max-imum and minimum values of objective functions, respectively,and these values are determined by solving the original model assingle-objective models.

To clear the steps of the proposed model, a flowchart is depictedin Fig. 4.

4. Application of the model with environmental criteria

In this section, the proposed integrated model is applied to sup-plier evaluation and order allocation with respect to environmen-tal criteria in a tissue paper manufacturing company which iscalled XYZ in this paper. XYZ produces different types of tissuepaper including facial tissues, paper towels, table napkins, etc.The company employs more than 160 people including managersand technical workers. The production capacity of XYZ is approxi-mately 5000 tons per annum, and the total net sales revenue of thecompany was about $31 million in 2016. XYZ has markets in morethan 30 countries including 62 customers (27 retailers and 35independent distributers). It has been certified to many standardslike ISO 9001:2015, ISO 10002:2014 and ISO 10668. Because of thepossible effect of the paper products on environment, XYZ hasbegun a plan since 2015 to improve its processes with respect toenvironmental aspects and get ISO 14001 certification. This com-pany purchases its raw material annually, and the tonnage of theraw material is approximately equal to the production capacity.As a part of its environmental plan, XYZ decided to supply theneeded raw material of a year from some suppliers according toenvironmental criteria. In the past 10 years, the company has pur-chased the raw material from more than 14 suppliers, but some ofthem have not met the required quality of XYZ. Therefore, the com-pany decided to limit the number of suppliers and choose supplierswhich have higher rates of quality, lower rates of lateness andadherence to environmental principles. For this aim, the board ofdirectors of the company formed a group of five experts (Dm1 to

Fig. 4. Flowchart of the proposed MCDM model.

Table 1The environmental evaluation criteria and their definitions.

Criteria Definition

C1 Environmental pollution This criterion is related to the estimatedlevel of emission of air pollutants,harmful materials, solid wastes,emission of air pollutants waste water,which releases by a supplier in itsproduction process

C2 Resource consumption This criterion is related to the estimatedlevel of raw material consumption,energy consumption and waterconsumption during the process ofproduction

C3 Ecological innovation This criterion is related to thedevelopment of processes and productsthat can help to sustainabledevelopment using the commercialapplication of knowledge to reach director indirect ecological improvements

C4 Environmental managementsystem

This criterion is related to planning ofresources for developing, organizationalstructure and implementing policies forenvironmental protection. ISO 14000and ISO14001 are the most widely usedstandards in an environmentalmanagement system

C5 Commitment of managers toenvironmentalimprovements

This criterion is related to the directparticipation of the highest levelmanagers of a company to improveenvironmental management practicesand performance

C6 Using green technologies inproduction processes

This criterion is related to theapplication of environmental scienceand green electronic devices to model,monitor and conserve the naturalresources of the environment and tocontrol the negative effects to theenvironment

C7 Using green materials inproduction processes

This criterion is related to the estimatedlevel of using recyclable material in allprocesses of production of the firmincluding packaging, manufacturing, etc.


Dm5) from different departments (production, research and devel-opment, purchasing, marketing and human resource manage-ment). Afterwards, the proposed integrated model is applied tothis example as follows:

Step 1. First mission of the group of experts, which is defined asdecision-makers, was to identify potential suppliers and definesome environmental criteria. According the literature(Govindan et al., 2015b; Nielsen, Banaeian, Golinska, Mobli, &Omid, 2014), the decision-makers defined seven criteria (C1 toC7) which are represented in Table 1. Then they estimated theaverage defect rates and the average rates of lateness for eachsupplier and also the amount of raw material (total demand)according to the historical data of the company. The data ofthe company which was used to define the problem is providedin Table 2.Step 2. In this step, each decision-maker should express theimportance of each criterion and also the performance valueof each supplier with respect to each criterion. The linguisticvariables which are defined based on TIT2FSs are used bydecision-makers to express their evaluations. The linguisticvariables and the corresponding TIT2FSs are presented inTable 3.

The importance of each criterion and ratings of alternativesexpressed by each decision-maker are shown in Tables 4 and 5,respectively.

Step 3. The aggregated weights of the criteria and performancevalues of the suppliers (elements of the decision matrix) are cal-culated using Eqs. (27) and (28). The results are represented inTables 6 and 7.Step 4. According to Table 7 and Eq. (29), we can calculate theelements of the average solution. Table 8 presents the TIT2FSsrelated to the elements of the average solution.Step 5. The values of positive and negative distances from theaverage solution are calculated using Eqs. (30)–(32). The

Table 2The data of the company related to the problem.

Year Average

2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Amount of raw material (ton) 4633 5109 4701 5472 4600 5428 4726 4914 5944 4069 4959.6

Purchasing cost (�102 $/ton) S1 3.85 3.8 4.3 4.45 4 4.8 4.9 4.8 4.9 5.1 4.49S2 5.5 5.9 6.3 6.2 6.2 6.4 6.6 6.5 6.7 7.1 6.34S3 5.7 5.8 5.7 6 6.1 6.1 6.2 6.4 6.5 6.9 6.14S4 2.5 2.6 2.7 2.9 3.2 3.4 3.5 3.5 3.6 3.6 3.15S5 5.2 5.4 5.3 5.65 5.5 5.8 6 6.2 6.1 6.45 5.76

Defect rate (%) S1 0.30 0.19 0.24 0.26 0.30 0.21 0.28 0.21 0.30 0.25 0.25S2 0.25 0.18 0.15 0.22 0.19 0.22 0.30 0.19 0.22 0.10 0.20S3 0.10 0.11 0.10 0.11 0.08 0.14 0.10 0.09 0.09 0.10 0.10S4 0.29 0.33 0.38 0.35 0.31 0.38 0.45 0.31 0.35 0.33 0.35S5 0.18 0.18 0.16 0.10 0.19 0.14 0.17 0.12 0.17 0.12 0.15

Rate of lateness (%) S1 0.10 0.06 0.12 0.11 0.09 0.11 0.10 0.11 0.11 0.05 0.10S2 0.27 0.24 0.24 0.26 0.22 0.27 0.26 0.21 0.30 0.20 0.25S3 0.23 0.25 0.28 0.10 0.24 0.32 0.25 0.25 0.26 0.28 0.25S4 0.17 0.13 0.13 0.19 0.12 0.16 0.18 0.16 0.15 0.12 0.15S5 0.08 0.08 0.09 0.10 0.12 0.09 0.09 0.10 0.11 0.12 0.10

Capacity (ton) S1 1200 1400 1100 1700 1200 1000 1000 1200 1100 1100 1200S2 1000 1500 1700 1600 1500 1400 1400 1700 1600 1600 1500S3 1200 2300 1500 1800 1200 2000 1900 1500 1700 1900 1700S4 1100 1350 1650 1500 1000 1200 1800 1700 1200 1500 1400S5 1000 1100 1250 1350 1600 1800 1700 1800 1800 1600 1500

Minimum allowable order (ton) S1 50 50 50 100 150 200 100 100 100 100 100S2 100 100 200 200 150 150 150 150 150 150 150S3 150 150 150 100 100 200 200 150 150 150 150S4 100 100 100 100 100 100 100 100 100 100 100S5 50 50 50 150 150 150 100 100 100 100 100

Table 3Linguistic variables and the corresponding TIT2FSs.

Usage Linguistic variable TIT2FSs

For weighting criteria Very low (VL) ((0,0,0,0.1;1,1),(0,0,0,0.05;0.9,0.9))Low (L) ((0,0.1,0.15,0.3;1,1),(0.05,0.1,0.15,0.2;0.9,0.9))Medium low (ML) ((0.1,0.3,0.35,0.5;1,1),(0.2,0.3,0.35,0.4;0.9,0.9))Medium (M) ((0.3,0.5,0.55,0.7;1,1),(0.4,0.5,0.55,0.6;0.9,0.9))Medium high (MH) ((0.5,0.7,0.75,0.9;1,1),(0.6,0.7,0.75,0.8;0.9,0.9))High (H) ((0.7,0.85,0.9,1;1,1),(0.8,0.85,0.9,0.95;0.9,0.9))Very high (VH) ((0.9,1,1,1;1,1),(0.95,1,1,1;0.9,0.9))

For rating alternatives Very poor (VP) ((0,0,0,1;1,1),(0,0,0,0.5;0.9,0.9))Poor (P) ((0,1,1.5,3;1,1),(0.5,1,1.5,2;0.9,0.9))Medium poor (MP) ((1,3,3.5,5;1,1),(2,3,3.5,4;0.9,0.9))Fair (F) ((3,5,5.5,7;1,1),(4,5,5.5,6;0.9,0.9))Medium good (MG) ((5,7,7.5,9;1,1),(6,7,7.5,8;0.9,0.9))Good (G) ((7,8.5,9,10;1,1),(8,8.5,9,9.5;0.9,0.9))Very good (VG) ((9,10,10,10;1,1),(9.5,10,10,10;0.9,0.9))

Table 4The importance of the criteria by each decision-maker.

Criteria Decision-makers

Dm1 Dm2 Dm3 Dm4 Dm5

C1 VH H VH MH HC2 H H M H MC3 L M ML M MC4 H M H H MC5 L ML ML ML VLC6 ML MH MH MH MLC7 M MH MH MH M


TIT2FSs related to the positive and negative distances are pre-sented in Tables 9 and 10, respectively.Step 6. The weighted sums of positive and negative distancesfor each supplier are determined according to Eqs. (33) and

(34). Here, we can present the TIT2FSs related to the weighted

sums. Tables 11 and 12 represent the elements of ggSP i andggNP i of suppliers, respectively.

Table 5The performance values of the suppliers by each decision-maker.

Decision-makers Suppliers Criteria

C1 C2 C3 C4 C5 C6 C7

Dm1 S1 F MG F F MP P MPS2 G VG MG MG MG MP MGS3 VG F MG G MP F FS4 P MP F P MG P MPS5 G MG G MG P MP MP

Dm2 S1 F MP G P MG P PS2 F G G MG G MP FS3 VG F MG VG F MG MPS4 VP MP MG F MP MP MPS5 F F F G P F MP

Dm3 S1 MP MP MG F MG P PS2 F VG MG G MG F MPS3 G F MG MG MG G MGS4 VP MP MP F F F PS5 F MP F G MP F MP

Dm4 S1 F F MG MP F VP PS2 F MG MG MG F F MGS3 G MG F VG F F MGS4 VP P F P MP F FS5 MG MP F MG F MP MP

Dm5 S1 P F G MP F P PS2 G MG VG F MG MP MGS3 G G MG G F MG MPS4 P MP MG F F MP FS5 MG F G G MP F F

Table 6The aggregated weights of the criteria.

~wjL ~wjU

wjL1 wjL2 wjL3 wjL4 hwj

L1 hwj

L2wjU1 wjU2 wjU3 wjU4 hwj

U1 hwj

U2eew10.74 0.88 0.91 0.98 1 1 0.82 0.88 0.91 0.94 0.9 0.9eew20.54 0.71 0.76 0.88 1 1 0.64 0.71 0.76 0.81 0.9 0.9eew30.2 0.38 0.43 0.58 1 1 0.29 0.38 0.43 0.48 0.9 0.9eew40.54 0.71 0.76 0.88 1 1 0.64 0.71 0.76 0.81 0.9 0.9eew50.06 0.2 0.24 0.38 1 1 0.13 0.2 0.24 0.29 0.9 0.9eew60.34 0.54 0.59 0.74 1 1 0.44 0.54 0.59 0.64 0.9 0.9eew70.42 0.62 0.67 0.82 1 1 0.52 0.62 0.67 0.72 0.9 0.9


Steps 7 and 8. The ranking method, which proposed in themethodology section (Eqs. (10)–(15)), is used in this step to cal-culate the positive and negative scores of each supplier withrespect to the considered environmental criteria. These valuesand the other parameters of the problem, which are used to for-mulate the multi-objective model, are provided in Table 13.

According to the parameters provided in Table 13 and Eqs. (35)–(43), the following multi-objective linear programming model isformulated.

Max Z1 ¼ 0:1616x1 þ 0:2369x2 þ 0:2569x3 þ 0:1402x4 þ 0:2044x5

Min Z2 ¼ 0:2519x1 þ 0:1483x2 þ 0:1541x3 þ 0:2548x4 þ 0:1910x5

Min Z3 ¼ 4:5x1 þ 6:3x2 þ 6:1x3 þ 3:1x4 þ 5:8x5

Subject to:

x1 þ x2 þ x3 þ x4 þ x5 ¼ 5000

0:25x1 þ 0:2x2 þ 0:1x3 þ 0:35x4 þ 0:15x5 6 12

0:1x1 þ 0:25x2 þ 0:25x3 þ 0:15x4 þ 0:1x5 6 18

x1 P 100y1; x2 P 150y2; x3 P 150y3; x4 P 100y4; x5 P 100y5

x1 6 1200y1; x2 6 1500y2; x3 6 1700y3; x4 6 1400y4; x5 6 1500y5

x1; x2; x3; x4; x5 P 0

y1; y2; y3; y4; y5 2 f0;1g

Step 9. By solving the formulated model of the previous step assingle-objective models, we first determine the maximum andminimum values of each objective separately. These values

are: Zmax1 = 1147.167, Zmin

1 = 909.954, Zmax2 = 1084.057,

Zmin2 = 846.406, Zmax

3 = 29870 and Zmin3 = 23930. Then the multi-

objective model is transformed to the following single-objective model according to Eqs. (44)–(50).

Max k

Subject to:

k 6 1þ Z1 � 1147:167237:213

k 6 1� Z2 � 846:406237:651

Table 7The aggregated performance values of the suppliers.

~xijL ~xijU

xijL1 xijL2 xijL3 xijL4 hxijL1 hxij

L2xijU1 xijU2 xijU3 xijU4 hxij

U1 hxijU2eex11

2 3.8 4.3 5.8 1 1 2.9 3.8 4.3 4.8 0.9 0.9eex122.6 4.6 5.1 6.6 1 1 3.6 4.6 5.1 5.6 0.9 0.9eex135.4 7.2 7.7 9 1 1 6.4 7.2 7.7 8.2 0.9 0.9eex141.6 3.4 3.9 5.4 1 1 2.5 3.4 3.9 4.4 0.9 0.9eex153.4 5.4 5.9 7.4 1 1 4.4 5.4 5.9 6.4 0.9 0.9eex160 0.8 1.2 2.6 1 1 0.4 0.8 1.2 1.7 0.9 0.9eex170.2 1.4 1.9 3.4 1 1 0.8 1.4 1.9 2.4 0.9 0.9eex214.6 6.4 6.9 8.2 1 1 5.6 6.4 6.9 7.4 0.9 0.9eex227 8.5 8.8 9.6 1 1 7.8 8.5 8.8 9.1 0.9 0.9eex236.2 7.9 8.3 9.4 1 1 7.1 7.9 8.3 8.7 0.9 0.9eex245 6.9 7.4 8.8 1 1 6 6.9 7.4 7.9 0.9 0.9eex255 6.9 7.4 8.8 1 1 6 6.9 7.4 7.9 0.9 0.9eex261.8 3.8 4.3 5.8 1 1 2.8 3.8 4.3 4.8 0.9 0.9eex273.8 5.8 6.3 7.8 1 1 4.8 5.8 6.3 6.8 0.9 0.9eex317.8 9.1 9.4 10 1 1 8.6 9.1 9.4 9.7 0.9 0.9eex324.2 6.1 6.6 8 1 1 5.2 6.1 6.6 7.1 0.9 0.9eex334.6 6.6 7.1 8.6 1 1 5.6 6.6 7.1 7.6 0.9 0.9eex347.4 8.8 9.1 9.8 1 1 8.2 8.8 9.1 9.4 0.9 0.9eex353 5 5.5 7 1 1 4 5 5.5 6 0.9 0.9eex364.6 6.5 7 8.4 1 1 5.6 6.5 7 7.5 0.9 0.9eex373 5 5.5 7 1 1 4 5 5.5 6 0.9 0.9eex410 0.4 0.6 1.8 1 1 0.2 0.4 0.6 1.1 0.9 0.9eex420.8 2.6 3.1 4.6 1 1 1.7 2.6 3.1 3.6 0.9 0.9eex433.4 5.4 5.9 7.4 1 1 4.4 5.4 5.9 6.4 0.9 0.9eex441.8 3.4 3.9 5.4 1 1 2.6 3.4 3.9 4.4 0.9 0.9eex452.6 4.6 5.1 6.6 1 1 3.6 4.6 5.1 5.6 0.9 0.9eex461.6 3.4 3.9 5.4 1 1 2.5 3.4 3.9 4.4 0.9 0.9eex471.6 3.4 3.9 5.4 1 1 2.5 3.4 3.9 4.4 0.9 0.9eex514.6 6.5 7 8.4 1 1 5.6 6.5 7 7.5 0.9 0.9eex522.6 4.6 5.1 6.6 1 1 3.6 4.6 5.1 5.6 0.9 0.9eex534.6 6.4 6.9 8.2 1 1 5.6 6.4 6.9 7.4 0.9 0.9eex546.2 7.9 8.4 9.6 1 1 7.2 7.9 8.4 8.9 0.9 0.9eex551 2.6 3.1 4.6 1 1 1.8 2.6 3.1 3.6 0.9 0.9eex562.2 4.2 4.7 6.2 1 1 3.2 4.2 4.7 5.2 0.9 0.9eex571.4 3.4 3.9 5.4 1 1 2.4 3.4 3.9 4.4 0.9 0.9

Table 8The elements of the average solution.

~V jL~V jU

V jL1 VjL2 V jL3 VjL4 hV j

L1 hV j

L2V jU1 V jU2 V jU3 V jU4 hV j

U1 hV j

U2eeV 13.8 5.24 5.64 6.84 1 1 4.58 5.24 5.64 6.1 0.9 0.9eeV 23.44 5.28 5.74 7.08 1 1 4.38 5.28 5.74 6.2 0.9 0.9eeV 34.84 6.7 7.18 8.52 1 1 5.82 6.7 7.18 7.66 0.9 0.9eeV 44.4 6.08 6.54 7.8 1 1 5.3 6.08 6.54 7 0.9 0.9eeV 53 4.9 5.4 6.88 1 1 3.96 4.9 5.4 5.9 0.9 0.9eeV 62.04 3.74 4.22 5.68 1 1 2.9 3.74 4.22 4.72 0.9 0.9eeV 72 3.8 4.3 5.8 1 1 2.9 3.8 4.3 4.8 0.9 0.9


k 6 1� Z3 � 239305940

Z1 ¼ 0:1616x1 þ 0:2369x2 þ 0:2569x3 þ 0:1402x4 þ 0:2044x5

Z2 ¼ 0:2519x1 þ 0:1483x2 þ 0:1541x3 þ 0:2548x4 þ 0:1910x5Z3 ¼ 4:5x1 þ 6:3x2 þ 6:1x3 þ 3:1x4 þ 5:8x5

x1 þ x2 þ x3 þ x4 þ x5 ¼ 5000

0:25x1 þ 0:2x2 þ 0:1x3 þ 0:35x4 þ 0:15x5 6 12

0:1x1 þ 0:25x2 þ 0:25x3 þ 0:15x4 þ 0:1x5 6 18

x1 P 100y1; x2 P 150y2; x3 P 150y3; x4 P 100y4; x5 P 100y5x1 6 1200y1; x2 6 1500y2; x3 6 1700y3; x4 6 1400y4; x5 6 1500y5x1; x2; x3; x4; x5 P 0

y1; y2; y3; y4; y5 2 f0;1g

Table 9The TIT2FSs related to the positive distances.

gPdijLgPdijU

PdijL1 PdijL2 PdijL3 PdijL4 hPdij

L1 hPdij

L2PdijU1 PdijU2 PdijU3 PdijU4 hPdij

U1 hPdij

U2ggPd110 0 0 0 1 1 0 0 0 0 1 1

ggPd120 0 0 0 1 1 0 0 0 0 1 1

ggPd13�0.4547 0.0029 0.1457 0.6062 1 1 �0.1836 0.0029 0.1457 0.3468 0.9 0.9

ggPd140 0 0 0 1 1 0 0 0 0 1 1

ggPd15�0.6854 0.0000 0.1970 0.8666 1 1 �0.2954 0.0000 0.1970 0.4806 0.9 0.9

ggPd160 0 0 0 1 1 0 0 0 0 1 1

ggPd170 0 0 0 1 1 0 0 0 0 1 1

ggPd21�0.4146 0.1407 0.3073 0.8144 1 1 �0.0925 0.1407 0.3073 0.5220 0.9 0.9

ggPd22�0.0147 0.5082 0.6482 1.1344 1 1 0.2946 0.5082 0.6482 0.8692 0.9 0.9

ggPd23�0.3381 0.1049 0.2332 0.6645 1 1 �0.0816 0.1049 0.2332 0.4197 0.9 0.9

ggPd24�0.4482 0.0576 0.2113 0.7043 1 1 �0.1601 0.0576 0.2113 0.4162 0.9 0.9

ggPd25�0.3703 0.2954 0.4924 1.1424 1 1 0.0197 0.2954 0.4924 0.7760 0.9 0.9

ggPd26�0.9871 �0.1069 0.1425 0.9566 1 1 �0.4885 �0.1069 0.1425 0.4834 0.9 0.9

ggPd27�0.5012 0.3759 0.6265 1.4534 1 1 0.0000 0.3759 0.6265 0.9773 0.9 0.9

ggPd310.1777 0.6404 0.7700 1.1476 1 1 0.4627 0.6404 0.7700 0.9477 0.9 0.9

ggPd32�0.5303 0.0663 0.2431 0.8397 1 1 �0.1841 0.0663 0.2431 0.5009 0.9 0.9

ggPd330 0 0 0 1 1 0 0 0 0 1 1

ggPd34�0.0640 0.3617 0.4834 0.8644 1 1 0.1921 0.3617 0.4834 0.6563 0.9 0.9

ggPd35�0.7642 �0.0788 0.1182 0.7879 1 1 �0.3742 �0.0788 0.1182 0.4018 0.9 0.9

ggPd36�0.2748 0.5801 0.8294 1.6180 1 1 0.2239 0.5801 0.8294 1.1703 0.9 0.9

ggPd37�0.7017 0.1754 0.4260 1.2530 1 1 �0.2005 0.1754 0.4260 0.7768 0.9 0.9

ggPd410 0 0 0 1 1 0 0 0 0 1 1

ggPd420 0 0 0 1 1 0 0 0 0 1 1

ggPd430 0 0 0 1 1 0 0 0 0 1 1

ggPd440 0 0 0 1 1 0 0 0 0 1 1

ggPd450 0 0 0 1 1 0 0 0 0 1 1

ggPd460 0 0 0 1 1 0 0 0 0 1 1

ggPd470 0 0 0 1 1 0 0 0 0 1 1

ggPd51�0.4146 0.1592 0.3258 0.8514 1 1 �0.0925 0.1592 0.3258 0.5405 0.9 0.9

ggPd520 0 0 0 1 1 0 0 0 0 1 1

ggPd530 0 0 0 1 1 0 0 0 0 1 1

ggPd54�0.2561 0.2177 0.3714 0.8323 1 1 0.0320 0.2177 0.3714 0.5762 0.9 0.9

ggPd550 0 0 0 1 1 0 0 0 0 1 1

ggPd56�0.8853 �0.0051 0.2442 1.0583 1 1 �0.3867 �0.0051 0.2442 0.5851 0.9 0.9

ggPd570 0 0 0 1 1 0 0 0 0 1 1


The result of solving this model is presented in Table 14. Forcomparison, the result of solving a classic single-objective cost-based model which minimizes only the total purchasing cost (Z3)is also shown in this table.

As can be seen in Table 14, the value of total purchasing costobtained by optimization of the proposed model is greater than thatof the other model. However, the values of total positive and negativescores of suppliers are improved in the proposed model. Moreover, ifwe only minimize the total purchasing cost, the quantity of orderfrom suppliers can be significantly different from those determinedby the proposed model. In the current problem, we can see thatthe order quantity from second supplier will be equal to zero if thetotal purchasing cost is minimized. This is due to the good environ-mental scores of this supplier and the high purchasing cost of it.

5. Sensitivity analysis

In this section, we examine the effect of changing the weights ofenvironmental criteria on order quantity allocated to each supplier

and total cost of purchasing. The analysis is made for each of thecriteria separately. For this aim, the weight of a criterion, whichis considered for the analysis, is changed, and equal weights aregiven to the other criteria. To clear the analysis process, supposethat we want to examine effect of changing the weight of gth cri-terion (g 2 f1;2; . . . ;mg). Then we set the weight of this criterion towg , and the other criteria weights are calculated as follows:

wj ¼1�wg

m� 1j 2 f1;2; . . . ;mg and j– g ð51Þ

For example, in the problem of this study, if we setw3 to 0.3, theweights of the other criteria (w1, w2, w4, w5, w6 and w7) will beequal to 0.1167 1�0:3

7�1

� �. By changing the value of wg (increasing or

decreasing), we can get the effect of gth criterion on solution.Here we choose nine values (from 0.1 to 0.9) for the weight of

each environmental criterion and analyze the effect of variationof the weights. To perform this analysis, the proposed model issolved 63 (9 � 7) times.

Table 10The TIT2FSs related to the negative distances.

gNdijLgNdijU

NdijL1 NdijL2 NdijL3 NdijL4 hNdij

L1 hNdij

L2NdijU1 NdijU2 NdijU3 NdijU4 hNdij

U1 hNdij

U2ggNd11�0.3702 0.1740 0.3406 0.8958 1 1 �0.0407 0.1740 0.3406 0.5923 0.9 0.9

ggNd12�0.5819 0.0331 0.2099 0.8250 1 1 �0.2247 0.0331 0.2099 0.4788 0.9 0.9

ggNd130 0 0 0 1 1 0 0 0 0 1 1

ggNd14�0.1601 0.3489 0.5026 0.9924 1 1 0.1441 0.3489 0.5026 0.7203 0.9 0.9

ggNd150 0 0 0 1 1 0 0 0 0 1 1

ggNd16�0.1425 0.6462 0.8701 1.4450 1 1 0.3053 0.6462 0.8701 1.0990 0.9 0.9

ggNd17�0.3508 0.4761 0.7267 1.4033 1 1 0.1253 0.4761 0.7267 1.0024 0.9 0.9

ggNd210 0 0 0 1 1 0 0 0 0 1 1

ggNd220 0 0 0 1 1 0 0 0 0 1 1

ggNd230 0 0 0 1 1 0 0 0 0 1 1

ggNd240 0 0 0 1 1 0 0 0 0 1 1

ggNd250 0 0 0 1 1 0 0 0 0 1 1

ggNd260 0 0 0 1 1 0 0 0 0 1 1

ggNd270 0 0 0 1 1 0 0 0 0 1 1

ggNd310 0 0 0 1 1 0 0 0 0 1 1

ggNd320 0 0 0 1 1 0 0 0 0 1 1

ggNd33�0.5479 �0.0583 0.0845 0.5712 1 1 �0.2594 �0.0583 0.0845 0.3002 0.9 0.9

ggNd340 0 0 0 1 1 0 0 0 0 1 1

ggNd350 0 0 0 1 1 0 0 0 0 1 1

ggNd360 0 0 0 1 1 0 0 0 0 1 1

ggNd370 0 0 0 1 1 0 0 0 0 1 1

ggNd410.3702 0.8588 0.9699 1.2660 1 1 0.6441 0.8588 0.9699 1.0920 0.9 0.9

ggNd42�0.2136 0.4014 0.5782 1.1565 1 1 0.1436 0.4014 0.5782 0.8287 0.9 0.9

ggNd43�0.3731 0.1166 0.2594 0.7461 1 1 �0.0845 0.1166 0.2594 0.4751 0.9 0.9

ggNd44�0.1601 0.3489 0.5026 0.9604 1 1 0.1441 0.3489 0.5026 0.7043 0.9 0.9

ggNd45�0.7091 �0.0394 0.1576 0.8430 1 1 �0.3230 �0.0394 0.1576 0.4530 0.9 0.9

ggNd46�0.8548 �0.0407 0.2086 1.0380 1 1 �0.3816 �0.0407 0.2086 0.5648 0.9 0.9

ggNd47�0.8520 �0.0251 0.2255 1.0525 1 1 �0.3759 �0.0251 0.2255 0.5764 0.9 0.9

ggNd510 0 0 0 1 1 0 0 0 0 1 1

ggNd52�0.5819 0.0331 0.2099 0.8250 1 1 �0.2247 0.0331 0.2099 0.4788 0.9 0.9

ggNd53�0.4896 �0.0291 0.1137 0.5712 1 1 �0.2302 �0.0291 0.1137 0.3002 0.9 0.9

ggNd540 0 0 0 1 1 0 0 0 0 1 1

ggNd55�0.3151 0.3545 0.5515 1.1581 1 1 0.0709 0.3545 0.5515 0.8076 0.9 0.9

ggNd560 0 0 0 1 1 0 0 0 0 1 1

ggNd57�0.8520 �0.0251 0.2255 1.1026 1 1 �0.3759 �0.0251 0.2255 0.6014 0.9 0.9

Table 11The weighted sum of positive distances from the average solution.

gSP iLgSP iU

SPiL1 SPiL2 SPiL3 SPiL4 hSPiL1 hSPi

L2SPiU1 SPiU2 SPiU3 SPiU4 hSPi

U1 hSPiU2ggSP 1

�0.52 0.00 0.11 0.68 1 1 �0.17 0.00 0.11 0.31 0.9 0.9

ggSP 2�2.29 0.79 1.66 5.14 1 1 �0.38 0.79 1.66 2.97 0.9 0.9

ggSP 3�1.46 1.27 2.06 5.15 1 1 0.20 1.27 2.06 3.25 0.9 0.9

ggSP 40.00 0.00 0.00 0.00 1 1 0.00 0.00 0.00 0.00 0.9 0.9

ggSP 5�1.29 0.29 0.72 2.35 1 1 �0.31 0.29 0.72 1.35 0.9 0.9


The effect of varying the weight of C1 (environmental pollution)on the order quantities is depicted in Fig. 5. As can be seen, increas-ing the weight of C1 leads to decrease in the order from the S1 (x1)

and increase in the order from the S5 (x5). Little variation can alsobe seen in the values of x4. However, the values of x2 and x3 is com-pletely stable.

Table 12The weighted sum of negative distances from the average solution.

gNP iLgNP iU

NPiL1 NPiL2 NPiL3 NPiL4 hNPiL1 hNPi

L2NPiU1 NPiU2 NPiU3 NPiU4 hNPi

U1 hNPiU2ggNP 1

�1.41 1.07 1.85 4.70 1 1 0.07 1.07 1.85 2.95 0.9 0.9

ggNP 20.00 0.00 0.00 0.00 1 1 0.00 0.00 0.00 0.00 0.9 0.9

ggNP 3�0.32 �0.03 0.04 0.33 1 1 �0.12 �0.03 0.04 0.14 0.9 0.9

ggNP �1.87 1.28 2.13 5.49 1 1 0.06 1.28 2.13 3.40 0.9 0.9

ggNP 5�1.61 0.07 0.49 2.40 1 1 �0.55 0.07 0.49 1.20 0.9 0.9

Table 13The parameters of the problem.

Parameters S1 S2 S3 S4 S5

RðggSP iÞ0.1616 0.2369 0.2569 0.1402 0.2044

RðggNP iÞ0.2519 0.1483 0.1541 0.2548 0.1910

CPi (�102 $/Ton) 4.5 6.3 6.1 3.1 5.8

pi (%) 0.25 0.2 0.1 0.35 0.15li (%) 0.1 0.25 0.25 0.15 0.1CAPi (Ton) 1200 1500 1700 1400 1500Qmini (Ton) 100 150 150 100 100Pmax 12 (0.24% of total demand)Lmax 18 (0.36% of total demand)

DT 5000

Table 14The optimization results.

Variables Minimization of totalpurchasing cost

Optimization of the proposedmodel

Z1 928.0232 1058.92Z2 1084.057 933.39Z3 23930 26139.91x1 1200 261.61x2 0 1500x3 900 1700x4 1400 1400x5 1500 138.39

Fig. 5. Effect of changing the weight of ‘‘environmental pollution”.

Fig. 6. Effect of changing the weight of ‘‘resource consumption”.

Fig. 7. Effect of changing the weight of ‘‘ecological innovation”.


Fig. 6 shows the effect of changing the weight of C2 (resourceconsumption). Maximum effect of this criterion can be seen inthe order quantity from S1 and S4 (x1 and x4). If the weight of thiscriterion is set to a high value, the value of x1 increases sharply.Also the order quantity from S2 and S4 decreases in high valuesof the weight of this criterion.

We can see in Fig. 7 the effect of changing the weight of C3 (eco-logical innovation). The effects of this criterion appear in high val-

ues of its weight. It can be seen that the weight of this criterionaffects the order quantity from S1 more than the order quantity

Fig. 8. Effect of changing the weight of ‘‘environmental management system”.

Fig. 9. Effect of changing the weight of ‘‘commitment of managers to environmen-tal improvements”.

Fig. 10. Effect of changing the weight of ‘‘using green technologies in productionprocesses”.

Fig. 11. Effect of changing the weight of ‘‘using green materials in productionprocesses”.

Fig. 12. Effect of changing weight of the criteria on total purchasing cost.


from the other suppliers. Moreover, high values of the weight of C3

decreases the values of x3 and x4.The effect of changing the weight of C4 (environmental manage-

ment system) is shown in Fig. 8. As can be seen in this figure, thequantity of order from S2 and S5 is very sensisitive to the weightof this criterion. Also we can see that increasing the weight of thiscriterion leads to reduction in the value of x1.

In Fig. 9, we can see the effect of changing the weight of C5

(commitment of managers to environmental improvements). Thevalues of x1, x3 and x4 are sensitive to changing the weight of thiscriterion. The domain of variation in the values of x1 is about 900within the range considered for the weight, and it shows that thequantity of order from S1 is the most sensitive variable when theweight of C5 is changed.

We represent the effect of changing the weights of C6 (usinggreen technologies in production processes) and C7 (using greenmaterials in production processes) in Figs. 10 and 11, respectively.As can be seen in these figures, these criteria have little effect onquantity of order purchased from the suppliers. However, the valueof x1 relatively increases by increasing the weights of C6 and C7.

The effect of changing the weights of each criterion on total pur-chasing cost is also depicted in Fig. 12. It can be seen that increasein the weights of C5 (commitment of managers to environmentalimprovements), C6 (using green technologies in production pro-cesses) and C7 (using green materials in production processes)leads to reduction in total purchasing cost. Among these criteria,C5 has more impact on the cost than C6 and C7. On the other hand,increasing the weights of C1 (environmental pollution) and C4

(environmental management system) increases the total purchas-ing cost. In this case, C1 has more effect than C4. Although changingthe weights of C2 (resource consumption) and C3 (ecological inno-vation) leads to variation in the total purchasing cost, the variationis not so great.

6. Discussion and future directions

Supplier evaluation and order allocation is an important issuefor companies which want to have long-term contracts with sup-pliers. Although economic factors are important in the process ofevaluation of suppliers, environmental attributes of suppliers canalso be essential for some companies with green strategies. In thispaper, we have proposed an integrated model which considersboth economic and environmental factors in the process of sup-plier evaluation and order allocation.

The proposed model consists of two main step including evalu-ation of suppliers with respect to environmental criteria and order


allocation according to this evaluation and economic objective. Theevaluation of suppliers with respect to environmental criteria hasbeen made by some experts (decision-makers) subjectively.Because of uncertainty of information in a subjective evaluation,the fuzzy sets theory has been used to model the opinions ofexperts. Linguistic variables defined by interval type-2 fuzzy setshave been utilized to quantify the experts’ opinions. This type offuzzy sets is very flexible to capture the uncertainty of informationbecause of its characteristics. Based on the experts’ opinions andsome steps of the EDAS method two parameters are defined foreach supplier with respect to considered environmental criteria:the positive score and negative score of supplier. According tothe environmental scores of the suppliers and economic parame-ters, a multi-objective linear programming has been formulated.By using this multi-objective model, we can determine thequantity of order from each supplier. In this study a fuzzy multi-objective programming approach has been used to determine thefinal solution. As shown in the sensitivity analysis section, bychanging the weight of each criterion using the proposed pattern,we can examine the effect of each environmental criterion on totalpurchasing cost and quantity of order from each supplier. How-

ever, we have to determine the values of Zmax1 , Zmin

1 , Zmax2 and Zmin

2

if the fuzzy multi-objective programming approach is used to solvethe model. According to the above-mentioned discussion, the pro-posed model has two main advantages as follows:

� Using IT2FSs helps us to consider more degrees of uncertainty inthe process of supplier evaluation.

� Using the EDAS method enables us to define environmental cri-teria as two scores (positive and negative) to determine thequantity of order from each supplier.

On the other hand, the proposed model has two main disadvan-tages as follows:

� Using IT2FSs in the evaluation process of suppliers requires aconsiderable amount of computations.

� Solving the proposed multi-objective model only yield one Par-eto optimal solution, while in such a problem we need to deter-mine the Pareto front (trade-off solutions).

The proposed model can provide us with a decision supportmechanism in the buyer–supplier relationship. We can have along-term buyer–supplier relationship if the evaluation processof suppliers is carried out correctly, and the proposed model helpsto optimize this process. Moreover, Lee and Klassen (2008) indi-cated that buyers’ supply chain management can initiate andenable the improvement of suppliers’ environmental capabilities.The buyer–supplier relationship can be established in the supplychain of many industries. Accordingly, if we can define appropriateenvironmental criteria for evaluation of suppliers in an industry,the proposed model can be applicable in that industry. However,the proposed model can only be used in a single-product andsingle-period supply chain. The application area of the proposedmodel includes many manufacturing industries such as automo-tive industry, food industry, clothing industry, and pulp and paperindustry.

Sustainable development or sustainability has been describedregarding three dimensions, i.e. economic, environmental andsocial. Each of these dimensions includes many criteria and sub-criteria which can significantly affect the competitive environmentof companies (Manea, Titan, Boboc, & Anoaica, 2016; Popescu,Boboc, Stoian, Zaharia, & Ladaru, 2017). These criteria and sub-criteria may be changed over time. The proposed model has beenestablished based on the economic and environmental dimensions

and can be extended to include the social dimension. However, thechangeable feature of the criteria and sub-criteria related to thesustainability dimensions requires flexibility in the proposedmodel for identification of evaluation criteria. This flexibilitydepends on the level of expertise of the decision-makers.

Based on the proposed model, we can give some recommenda-tions for future research. To determine the positive and negativescores of the suppliers with respect to environmental criteria, wehave used a ranking method of interval type-2 fuzzy sets whichproposed by Keshavarz Ghorabaee et al. (2016a). The other rankingmethods can also be used in this step in future research. The pro-posed multi-objective model can also yield the trade-off solutionswith respect to environmental and economic objectives. Futureresearch can apply the other multi-objective programming meth-ods like e-constraint method to obtain the trade-off solutions.Moreover, the proposed model is formulated for a single-productand single-period situation. Future research can also extend themodel to a multi-product and multi-period model.

7. Conclusion

In this paper we have proposed an integrated model for supplierevaluation and order allocation with respect to environmental cri-teria. Based on the EDAS method and interval type-2 fuzzy sets, aprocess has been developed to evaluate suppliers and determinepositive and negative scores of each supplier. Then a multi-objective linear programming model has been formulated fordetermination of order quantity from each supplier that maximizestotal positive score and minimizes total negative score and totalpurchasing cost. By using a fuzzy multi-objective programmingapproach an auxiliary model has been presented to solve the pro-posed multi-objective model. The proposed integrated model hasbeen applied to a numerical example of supplier evaluation andorder allocation with five suppliers and seven environmental crite-ria, and the optimal solution has been determined. Also we haveperformed a sensitivity analysis to examine the effect of variationin the weights of the environmental criteria on the quantity oforder and total purchasing cost. The results of this analysis showthat ‘‘resource consumption,” ‘‘ecological innovation,” ‘‘environ-mental management system” and ‘‘commitment of managers toenvironmental improvements” are four criteria that have moreimpact on the order quantity than the other criteria. More overthe effects of ‘‘environmental pollution” and ‘‘commitment of man-agers to environmental improvements” on total purchasing costare more than the other environmental criteria for the companyconsidered in this study.

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